Finite Math and Applied Calculus (6th Edition)

Published by Brooks Cole
ISBN 10: 1133607705
ISBN 13: 978-1-13360-770-0

Chapter 1 - Section 1.4 - Linear Regression - Exercises - Page 102: 13a

Answer

$r=0.9958$, best, not perfect.

Work Step by Step

Regression line: $y=mx+b,$ $m=\displaystyle \frac{n(\sum xy)-(\sum x)(\sum y)}{n(\sum x^{2})-(\sum x)^{2}}\qquad b=\frac{\sum y-m(\sum x)}{n}$ $n=$ number of data points. Correlation Coefficient: $r=\displaystyle \frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{n(\sum x^{2})-(\sum x)^{2}}\cdot\sqrt{n(\sum y^{2})-(\sum y)^{2}}}$ Interpretation If $r$ is positive, the regression line has positive slope; if $r$ is negative, the regression line has negative slope. If $r=1$ or $- 1$, then all the data points lie exactly on the regression line; if it is close to $\pm 1$, then all the data points are close to the regression line. If $r$ is close to $0$, then $y$ does not depend linearly on $x.$ ----- (a) Using Excel, we set up a table \begin{array}{|c|c|c|c|c|c} & x & y & xy & xx & yy \\ \hline & 1 & 3 & 3 & 1 & 9 \\ & 2 & 4 & 8 & 4 & 16 \\ & 5 & 6 & 30 & 25 & 36 \\\hline \Sigma & {\bf 8} & {\bf 13} & {\bf 41} & {\bf 30} & {\bf 61} \\\hline points & 3 & & 123 & 90 & 183 \\ & & & & & \\ m= & 0.730769 & & & & \\ b= & 2.384615 & & & & \\ & & & r= & 0.995871 & \\ \end{array} m,b, and r are calculated according to the above formulas. Regression line:$\qquad y=0.7308x+2.3846,$ Correlation Coefficient: $\qquad r=0.9958$, which after calculating (b) and (c), turns out to be the best of the three
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